61.23.11 problem 11

Internal problem ID [12333]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.2.
Problem number : 11
Date solved : Monday, March 31, 2025 at 05:10:50 AM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }&=a \cos \left (\lambda x \right ) y+1 \end{align*}

✗ Maple

ode:=y(x)*diff(y(x),x) = a*cos(lambda*x)*y(x)+1; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*D[y[x],x]==a*Cos[\[Lambda]*x]*y[x]+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(-a*y(x)*cos(lambda_*x) + y(x)*Derivative(y(x), x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*cos(lambda_*x) + Derivative(y(x), x) - 1/y(x) cannot be solved by the factorable group method